Decomposition of the diagonal map

This paper presents a new method for using cup product information to draw conclusions about the Lusternik–Schnirelmann category of a space. The key idea is that of the Hopf set in <f>X</f> of a map <f>f : Sn−1→L</f>; if <f>K=L∪fDn⊆X</f>, then <f>catX(K)=catX(L)</f> if and only if <f>&ast;</f> is in the Hopf set in <f>X</f> of <f>f</f>. The main result explicitly constructs elements of the Hopf set in <f>X</f> of <f>f</f> in terms of members of the Hopf set in <f>X</f> of the attaching maps of lower dimensional cells. Applications include: a calculation of the category of <f>Sp(2)</f> without higher order cohomology operations; new, easily used upper bounds for Lusternik–Schnirelmann category that apply to any space; and new information about the category of the CW skeleta of loop spaces and free loop spaces on even-dimensional spheres. [Copyright &y& Elsevier]